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Algebra 2 2-2: Linear Equations Objective 1: I can graph a linear equation in slope-intercept or standard form. A function whose graph is a line is a ____________________. You can represent a linear function with a ____________________, such as __________. A solution of a linear equation is any __________________ (x, y) that makes the equation true. An equation in the form y mx b is said to be in ____________________________. Recall from algebra 1 that there are different names for the variables x and y: x y_______________________ Let’s review the steps to graph a linear equation that you learned in algebra and geometry. * Plot _____ on the y-axis. If b is negative, plot it on the _______________ axis, if b is positive plot it on the ________________ axis. rise . run If the slope is positive, move _________________________________________ and if the If the slope is negative, move _________________________________________ Use the slope to get more points on the line. Recall that the slope, m Example 1: Graph each equation. 2 a) y x 3 3 1 b) y x 5 QC1: Graph each equation. a) y x 3 b) y 2 x 3 Sometimes the equations are written as Ax By C , which is called _______________ ___________. There are two ways to graph an equation that is written in standard form. 1) Put it into slope-intercept form, y mx b . 2) Find the x and y-intercepts of the graph. We do this by letting x and y equal _______. The ____________________ is where the graph crosses the y-axis, at (0, b). The ____________________ is where the graph crosses the x-axis, at (a, 0) Example 2: Graph by finding the intercepts: x y 2 QC2: Graph by finding the intercepts: 3x 2 y 6 QC 2.5: Graph by finding the intercepts: 4 x 2 y 8 In the next objective, we are going to recall how to write equations of lines. To do so, we will need to remember how to use the slope formula. Slope = rise = run Example 3: Find the slope of the line through the points (3, 2) and (-9, 6). QC 3: Find the slope of the line through each pair of points. a) (-2, -2) and (4, 2) b) (0, -3) and (7, -9) It is important that you are able to quickly identify the slopes of any line. Chant: Vertical lines are __________________, horizontal ____________. Practice Problems: Find the slope of the line that contains each pair of points. 1. (3, -1) and (12, -2) 2. (-3, -5) and (7, 0) 3. (-4, 8) and (2, 2) 4. (6, -3) and (6, 1) 5. (3, -1) and (5, -1) Find the slope of the line in each graph. 7. 8. 9. 10. 6. (9, -3) and (7, 5) Objective 2: I can write equations of lines. To write an equation for a line, you need to be given 2 pieces of information. Case 1) or Case 2) Case 1: Writing an equation given the slope and a point, you need to use the _________________________: Example 4: Write an equation in standard form of the line with a slope of 1 that 2 contains the point (8, -1). QC 4: Write in standard form the equation of each line. A) Slope of 2, through (4, -2) 2 B) Slope of , through (5, 6) 3 Case 2: When you are given 2 points on a line, find the ________ first, then use the ______________________ from Case 1. Example 5: Write an equation in slope-intercept form for a line that passes through the points (1, 5) and (4, -1). QC5: Write in slope-intercept form the equation for each line that passes through each pair of points. A) (5, 0) and (-3, 2) B) (-4, -3) and (-5, 7) It is very important that you are able to rewrite an equation from standard form into slope-intercept form. We need this especially to use the graphing calculators. Ex 6: Rewrite 4 x 3 y 7 in slope-intercept form. Identify the slope and y-intercept. QC6: Rewrite 3x 2 y 1 in slope-intercept form. Identify the slope and y-intercept. Objective 3: I can determine whether lines are parallel or perpendicular. In the graph below, the two lines are ______________. Parallel lines are lines in the same plane that never _______________. The equation of the top line is: The equation of the bottom line is: What do these two equations have in common? Slopes of parallel lines: Non-vertical lines are parallel if they have the ____________________ and different ________________. Example: The equations and have the same slope, ____, and different y-intercepts. The graphs of the two equations are parallel. You can use _____________________ form of the equation of a line to determine whether the lines are parallel. 1 Example 7: Are the lines y x 5 and 2 x 6 y 12 parallel? Explain. 3 QC 7: Are the lines 6 x 8 y 24 and y 3 x 7 parallel? Explain. 4 The lines below are _______________________. Perpendicular lines are lines that intersect to form a ________________. The equation of the positive line is: The equation of the negative line is: What do you notice about the slopes of these lines? Slopes of Perpendicular Lines: Two lines are perpendicular if the product of their slopes is _____. Their slopes will also be ___________________________. A vertical and horizontal line are also perpendicular. Example: The equations and have opposite reciprocal slopes, (the product of the slopes is also -1), so these lines are perpendicular. Example 8: Are the lines for each pair of equations parallel, perpendicular, or neither? Explain. 1 y x4 A) 4 y 4x 2 2 x6 B) 3 2x 3 y 6 y C) 3x 5 y 3 5x 3 y 8 D) 2x y 5 y 2x 1 Example 9: Write an equation of a line that is parallel to y 3x 5 and goes through the point (-2, 1). Example 10: Write an equation of a line that is perpendicular to y 3x 5 and passes through the point (6, 2).